Last updated on May 26th, 2025
If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in fields such as vehicle design, finance, etc. Here, we will discuss the square root of 772.
The square root is the inverse of the square of a number. 772 is not a perfect square. The square root of 772 is expressed in both radical and exponential form. In the radical form, it is expressed as √772, whereas in exponential form, it is (772)^(1/2). √772 ≈ 27.78489, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.
The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers, where the long-division method and approximation method are used. Let us now learn the following methods:
The product of prime factors is the prime factorization of a number. Now let us look at how 772 is broken down into its prime factors.
Step 1: Finding the prime factors of 772 Breaking it down, we get 2 × 2 × 193: 2^2 × 193
Step 2: Since 772 is not a perfect square, the digits of the number can’t be grouped in pairs.
Therefore, calculating 772 using prime factorization alone cannot provide an exact square root.
The long division method is particularly used for non-perfect square numbers. Let's learn how to find the square root using the long division method, step by step.
Step 1: Group the digits of 772 from right to left. In this case, we have one group of two digits (72) and one group of one digit (7).
Step 2: Find the largest number whose square is less than or equal to 7. This number is 2, as 2^2 = 4. Subtract 4 from 7, leaving a remainder of 3.
Step 3: Bring down the next pair of digits (72) to get 372.
Step 4: Double the divisor (2) and find the next digit of the quotient such that (20 + n) × n is less than or equal to 372. The number is 7, as 27 × 7 = 189.
Step 5: Subtract 189 from 372 to get 183. Bring down two zeros to make it 18300.
Step 6: Repeat the process to find the next digit in the decimal places. Continue this process to find the square root to the desired precision.
Thus, √772 ≈ 27.78.
The approximation method is another method for finding square roots. Now let us learn how to approximate the square root of 772.
Step 1: Identify the perfect squares closest to 772.
The smallest perfect square less than 772 is 729, and the largest perfect square greater than 772 is 784. So, √772 falls between 27 and 28.
Step 2: Apply the formula: (Given number - smallest perfect square) / (largest perfect square - smallest perfect square).
Using this formula: (772 - 729) / (784 - 729) = 43 / 55 ≈ 0.7818. Adding the approximate decimal to the lower perfect square root gives 27 + 0.7818 ≈ 27.78.
Students often make mistakes while finding the square root, such as forgetting about the negative square root or skipping steps in the long division method. Let's look at some common mistakes and how to avoid them.
Can you help Max find the area of a square box if its side length is given as √772?
The area of the square is approximately 772 square units.
The area of the square = side^2.
The side length is given as √772.
Area of the square = (√772)² = 772.
Therefore, the area of the square box is approximately 772 square units.
A square-shaped building measuring 772 square feet is built; if each of the sides is √772, what will be the square feet of half of the building?
386 square feet
Divide the given area by 2 since the building is square-shaped.
Dividing 772 by 2 gives 386.
So half of the building measures 386 square feet.
Calculate √772 × 5.
Approximately 138.92
The first step is to find the square root of 772, which is approximately 27.78.
Multiply 27.78 by 5.
So, 27.78 × 5 ≈ 138.92.
What will be the square root of (772 + 28)?
The square root is 28.
To find the square root, calculate the sum of (772 + 28). 772 + 28 = 800, and then √800 = ±28.284271 (approximately ±28).
Therefore, the approximate square root of (772 + 28) is ±28.
Find the perimeter of the rectangle if its length ‘l’ is √772 units and the width ‘w’ is 50 units.
The perimeter of the rectangle is approximately 155.56 units.
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√772 + 50) ≈ 2 × (27.78 + 50) = 2 × 77.78 ≈ 155.56 units.
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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